Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Definition

Let "C" be a category and let {"X_i" | "i" ∈ "I"} be an indexed family of (not necessarily distinct) objects in "C". The product of the set {"X_i"} is an object "X" together with a collection of morphisms "π_i" : "X" → "X_i" (called the "canonical projections" or "projection morphisms", which are often, but not always, epimorphisms) which satisfy a universal property: for any object "Y" and any collection of morphisms "f_i" : "Y" → "X_i", there exists a unique morphism "f" : "Y" → "X" such that for all "i" ∈ "I" it is the case that "f_i" = "π_i" "f". That is, the following diagram commutes (for all "i"):

If the family of objects consists of only two members the product is usually written "X"₁×"X"₂, and the diagram takes the form:

The unique arrow "f" making this diagram commute is sometimes denoted ⟨"f"₁,"f"₂⟩.

Examples

* In the category Set (the category of sets), the product in the category theoretic sense is the cartesian product. Given a family of sets "X_i" the product is defined as:$prod\_\{i\; in\; I\}\; X\_i\; :=\; \{(x\_i)\_\{i\; in\; I\}\; |\; x\_i\; in\; X\_i\; ,\; forall\; i\; in\; I\}$with the canonical projections:$pi\_j\; :\; prod\_\{i\; in\; I\}\; X\_i\; o\; X\_j\; mathrm\{\; ,\; \}\; quad\; pi\_j((x\_i)\_\{i\; in\; I\})\; :=\; x\_j$Given any set "Y" with a family of functions:$f\_i\; :\; Y\; o\; X\_i$the universal arrow "f" is defined as:$f:Y\; o\; prod\_\{i\; in\; I\}\; X\_i\; mathrm\{\; ,\; \}\; quad\; f(y)\; :=\; (f\_i(y))\_\{i\; in\; I\}$

*In the category of topological spaces, the product is the space whose underlying set is the cartesian product and which carries the product topology.

*In the category of modules over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication.

*In the category of groups, the product is given by the cartesian product with multiplication defined componentwise.

* In the category of algebraic varieties, the categorical product is given by the Segre embedding.

* In the category of semi-abelian monoids, the categorical product is given by the history monoid.

* A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).

Discussion

The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any functor from a discrete category to "C". Not every family {"X"_"i"} needs to have a product, but if it does, then the product is unique in a strong sense: if "π"_"i" : "X" → "X"_"i" and "π"’_"i" : "X"’ → "X"_"i" are two products of the family {"X"_"i"}, then (by the definition of products) there exists a unique isomorphism "f" : "X" → "X"’ such that "π"_"i" = "π"’_"i" "f" for each "i" in "I".

As with any universal property, the product can be understood as a universal morphism. Let Δ: "C" → "C"×"C" be the diagonal functor which assigns to each object "X" the ordered pair ("X","X") and to each morphism "f":"X" → "Y" the pair ("f","f"). Then the product "X"×"Y" in "C" is given by a universal morphism from the functor Δ to the object ("X","Y") in "C"×"C".

An empty product (i.e. "I" is the empty set) is the same as a terminal object in "C".

If "I" is a set such that all products for families indexed with "I" exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor "C"^"I" → "C". The product of the family {"X"_"i"} is then often denoted by ∏_"i" "X"_"i", and the maps π_"i" are known as the natural projections. We have a natural isomorphism :$operatorname\{Hom\}\_Cleft(Y,prod\_\{iin\; I\}X\_i\; ight)\; simeq\; prod\_\{iin\; I\}operatorname\{Hom\}\_C(Y,X\_i)$(where Hom_"C"("U","V") denotes the set of all morphisms from "U" to "V" in "C", the left product is the one in "C" and the right is the cartesian product of sets). Thus the covariant hom-functor takes products to products. This is a consequence of the fact that the hom-functor is continuous.

If "I" is a finite set, say "I" = {1,...,"n"}, then the product of objects "X"₁,...,"X"_"n" is often denoted by "X"₁×...×"X"_"n".Suppose all finite products exist in "C", product functors have been chosen as above, and 1 denotes the terminal object of "C" corresponding to the empty product. We then have natural isomorphisms:$X\; imes\; (Y\; imes\; Z)simeq\; (X\; imes\; Y)\; imes\; Zsimeq\; X\; imes\; Y\; imes\; Z$:$X\; imes\; 1\; simeq\; 1\; imes\; X\; simeq\; X$:$X\; imes\; Y\; simeq\; Y\; imes\; X$These properties are formally similar to those of a commutative monoid; a category with its finite products and terminal object constitutes a symmetric monoidal category.

Distributivity

In general, there is a canonical morphism "X"×"Y"+"X"×"Z" → "X"×("Y"+"Z"), where the plus sign here denotes the coproduct. To see this, note that we have various canonical projections and injections which fill out the diagram

The universal property for "X"×("Y"+"Z") then guarantees a unique morphism "X"×"Y"+"X"×"Z" → "X"×("Y"+"Z"). A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism:$X\; imes\; (Y\; +\; Z)simeq\; (X\; imes\; Y)+\; (X\; imes\; Z).$

ee also

* Coproduct – the dual of the product
* Limit and colimits
* Equalizer
* Inverse limit
* Cartesian closed category
* Categorical pullback